Syllabus and Course Description

This course is intented to be elementary in the sense that no
other upper-division course is needed as a prerequisite. Our book
is indeed quite elementary and concrete. On the other hand, it
requires readers' close attention and includes some exercises for
which cleverness and insight may be helpful. Even its proofs of
standard results are often quite novel. Unlike many recent books,
it does not emphasize elliptic
curves or cryptography.
It seems to avoid mention of continued
fractions. We will follow the book closely, omitting Chapter
4 if we can. Here is a brief description of what we might study
this semester:

Chapter 1: Fundamental properties of the set of integers, including
prime
factorization and common divisors.

Graduate Student Instructor

Aaron Greicius
of the math department will be participating in this course in various
ways: he'll be grading homework, holding office hours,
assisting with exam grading and taking over occasional lectures.
His office hours: Wednesday 9-11 in 834 Evans.

Homework

Homework assignments will be due on Wednesday each week. There will be a
short set of problems for the first week.

Problem 34 is equivalent to a noterious job interview question: A
student comes into a locker room and opens all the lockers. A
second student closes every second locker door, beginning with the
second one. A third student changes the status of every third
door, beginning with the third. (She closes the third door but
opens the sixth, which was closed by the second student.) A fourth
student changes every fourth door, beginning with the fourth.
This goes on until N students have gone through; N is the number
of lockers. At the end of the process, which lockers are open?

Assignment due September 13:

§ 1.20 39-44

Assignment due Friday, September 22:

§ 1.24 46-49

§ 1.25 50, 51, 52, 53

§ 1.27 57 (parts f-o)

Assignment due Friday, September 29:
Exercises 61-64 on page 36. Exercise 65 is interesting and is
recommended; consider it as an optional problem.

Assignment due Friday, October 6:

§ 2.6 3, 4, 5

§ 2.11 11, 12, 13

§ 2.17 15, 16

Assignment due Friday the 13th:

§ 2.22 21

§ 2.35 22, 23, 24, 25, 26

§ 2.36 27

Assignment due Friday the 20th:

If p is a prime > 3, show that
sum of the reciprocals of the first (p-1) positive integers,
when written as a fraction, has a numerator
divisible by p^2. For example,
1 + 1/2 + ... + 1/10 = 7381/2520 and 7381= 121*61.
[Hint: you've essentially done this problem on a previous assignment!]

Assignment due Monday, November 13:
Chapter 6, problems 1, 5 and 6 (both basically done in class, so don't
hand in), 7, 8, 9, 10. Also, suppose that 5 is a square mod p, where p
is a prime. Show that the Fibonacci number Fp-1 is divisible
by p. For example,
F10 = 55 is divisible by 11, and
F28= 317811 is 10959 times 29.
(Never mind the last problem; I did it in class.)

Grading and Examinations

Course grades
will be based on a composite
"total score" that is intended to weight
the course components roughly as follows:
midterm exams 15% each, homework
25%, final exam 45%. Please note the following exam dates and times:

Final examination, Saturday 5-8PM, December 16, 2006
in 60 Evans; happy
holidays!
[Questions and possible solutions, written by
Ribet.]
Results:
This was not your greatest showing, friends. The median score was
26 out of 45. There were only two scores over 40; scores ranged from
10 to 45.